A123963 Triangle T(n, k) = k^4 - n^4 + 2*k*n*(1 - k^2*n^2), read by rows.

0, -1, 0, -16, -27, -120, -81, -128, -485, -1440, -256, -375, -1248, -3607, -8160, -625, -864, -2589, -7264, -16329, -31200, -1296, -1715, -4712, -12843, -28640, -54611, -93240, -2401, -3072, -7845, -20800, -45993, -87456, -149197, -235200, -4096, -5103, -12240, -31615, -69312, -131391, -223888, -352815, -524160

Offset

0,4

Comments

A triangular sequence based on the omega(3) Jacobian Elliptic Modular equation.

Links

G. C. Greubel, Rows n = 0..100 of the triangle, flattened

Eric Weisstein's World of Mathematics, Modular Equation

Formula

T(n, k) = k^4 - n^4 + 2*k*n*(1 - k^2*n^2).

Sum_{k=0..n} T(n, k) = (-1/15)*binomial(n+1, 2) * (15*n^5 +15*n^4 +24*n^3 -9*n^2 -31*n +1). - G. C. Greubel, Feb 20 2021

Example

Triangular sequence:
      0;
     -1,     0;
    -16,   -27,   -120;
    -81,  -128,   -485,  -1440;
   -256,  -375,  -1248,  -3607,  -8160;
   -625,  -864,  -2589,  -7264, -16329,  -31200;
  -1296, -1715,  -4712, -12843, -28640,  -54611,  -93240;
  -2401, -3072,  -7845, -20800, -45993,  -87456, -149197, -235200;
  -4096, -5103, -12240, -31615, -69312, -131391, -223888, -352815, -524160;

Mathematica

T[n_, k_]:= k^4 - n^4 + 2*n*k*(1 - k^2*n^2);
Table[T[n, k], {n, 0, 12}, {k, 0, n}]//Flatten (* modified by G. C. Greubel, Feb 20 2021 *)

Prog

(Sage) flatten([[k^4 - n^4 + 2*n*k*(1 - k^2*n^2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Feb 20 2021
(Magma) [k^4 - n^4 + 2*n*k*(1 - k^2*n^2): k in [0..n], n in [0..12]]; // G. C. Greubel, Feb 20 2021

Crossrefs

Sequence in context: A329206 A280935 A067650 * A073396 A338093 A302553

Adjacent sequences: A123960 A123961 A123962 * A123964 A123965 A123966

Keyword

sign,tabl

Author

Roger L. Bagula, Oct 28 2006

Extensions

Edited by G. C. Greubel, Feb 20 2021

Status

approved